#140047

order	512
nilpotency class	4
isoclinism	isoclinic class of #139956
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#3493	C₂²
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

409 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9,6,10,4,7)(3,11,13,8,14,15,12,16)(17,18,22,27)(19,28,33,45)(20,34,29,25)(21,26)(23,30,41,31)(24,37)(32,42)(35,43)(36,39)(38,44)(40,47)(46,48);(2,7,10,9)(8,11,16,15)(17,19,29,30,22,33,20,31)(18,23,34,45,27,41,25,28)(21,36,37,47,38,32,35,46)(24,42,26,40,43,39,44,48);(1,3,6,14)(2,8,10,16)(4,12,5,13)(7,11,9,15)(17,20,22,29)(18,24,27,43)(19,30,33,31)(21,37,38,35)(23,39,41,42)(25,44,34,26)(28,40,45,48)(32,47,36,46)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=g,e^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=egh,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,e^c=eg,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47243);